Anyon Permutations in Quantum Double Models through Constant-depth Circuits

We provide explicit constant-depth local unitary circuits that realize general anyon permutations in Kitaev’s quantum double models. This construction can be naturally understood through a correspondence between anyon permutation symmetries of two-dimensional topological orders and self-dualities in one-dimensional systems, where local gates implement self-duality transformations on the boundaries of microscopic regions. From this holographic perspective, general anyon permutations in the $D(G)$ quantum double correspond to compositions of three classes of one-dimensional self-dualities, including gauging of certain subgroups of $G$, stacking with $G$ symmetry-protected topological phases, and outer automorphisms of the group $G$. We construct circuits realizing the first class by employing self-dual unitary gauging maps, and present transversal circuits for the latter two classes.

💡 Research Summary

This paper presents a systematic construction of constant‑depth local unitary circuits that implement arbitrary anyon permutations in Kitaev’s quantum double models (D(G)) for any finite group (G). The authors motivate the work by noting that logical operations in topological quantum memories are realized by braiding or permuting anyonic excitations, and that non‑Clifford gates (e.g., the T‑gate) require the ability to enact non‑trivial anyon automorphisms, which correspond to invertible domain walls in the underlying topological order. While previous studies have focused on specific groups or limited families of permutations, this work provides a unified, group‑theoretic framework that covers all possible permutations.

The central conceptual bridge is a holographic correspondence between 2‑dimensional anyon‑permutation symmetries and 1‑dimensional self‑duality transformations. In a 1‑D setting, a self‑duality is a locality‑preserving automorphism (a quantum cellular automaton, QCA) acting on the algebra of symmetry‑invariant operators. For Abelian groups it is known that any such QCA lifts to an anyon permutation in the associated 2‑D quantum double; the authors extend this to arbitrary finite groups by invoking the Brauer‑Picard (duality) group (G(G)), which is mathematically isomorphic to the group of braided auto‑equivalences of (D(G)).

The duality group (G(G)) is generated by three classes of 1‑D self‑dualities:

1. Gauging dualities – When (G) contains a normal Abelian subgroup (N) with quotient (Q=G/N), the group can be expressed as an extension ((n,q)) with a 2‑cocycle (\omega) and a conjugation action (\sigma). A self‑dual gauging map (\Gamma) is built from a bicharacter (\chi) on (N) that is invariant under (Q). The map acts locally as a sequence of Hadamard, phase (S), and controlled‑Z gates on the (N)‑part, while leaving the (Q)‑part untouched. When the extension does not split, a twisted version of the gauging map is constructed. This transformation implements a partial electric‑magnetic duality in (D(G)), exchanging pure fluxes labeled by (